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Network Models

Network Models. Please carefully study the following examples: Transportation problem - section 3.11 and 5.4 Transshipment problem - section 5.1 Shortest path problem - section 5.2 Maximal flow problem - section 5.6 Minimal spanning tree problem - section 5.8 Suggested problems:

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Network Models

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  1. Network Models • Please carefully study the following examples: • Transportation problem - section 3.11 and 5.4 • Transshipment problem - section 5.1 • Shortest path problem - section 5.2 • Maximal flow problem - section 5.6 • Minimal spanning tree problem - section 5.8 • Suggested problems: • 5th & 6thed: #14, 19, 20, 31 • Int’l ed: # 5, 26, 28, 4

  2. Transportation problem statement

  3. 21 miles Mt. Dora Ocala Capacity 1 4 50 275,000 200,000 40 35 30 Eustis Orlando 600,000 2 5 400,000 22 55 20 Clermont Leesburg 225,000 3 6 25 300,000 Transportation problem (recording available – zip…) Demand (processing plants) Supply

  4. Transportation Problem Formulation Objective & decision Variables? What kinds (sets) of constraints?

  5. The Bavarian Motor Company (BMC) (recording available) The Bavarian Motor Company ( BMC) manufactures expensive luxury cars in Hamburg, Germany, and exports cars to sell in the United States. The exported cars are shipped from Hamburg to ports in Newark, New Jersey, and Jacksonville, Florida. From these ports, the cars are transported by rail or truck to distributors located in Boston, Massachusetts; Columbus, Ohio; Atlanta, Georgia; Richmond, Virginia; and Mobile, Alabama. Figure shows the possible shipping routes avail-able to the company along with the transportation cost for shipping each car along the indicated path. Currently, 200 cars are available at the port in Newark and 300 are available in Jacksonville. The numbers of cars needed by the distributors in Boston, Columbus, Atlanta, Richmond, and Mobile are 100, 60, 170, 80, and 70, respectively. BMC wants to determine the least costly way of transporting cars from the ports in Newark and Jacksonville to the cities where they are needed.

  6. +100 Boston $30 $50 2 -200 Newark +60 1 Columbus $40 3 $40 $35 Richmond $30 Atlanta +80 4 +170 5 $25 $50 $45 $35 Mobile +70 J'ville -300 6 $50 7 Transshipment problem • Network flow problems can be represented as a collection of nodes connected by arcs. • There are three types of nodes: • Supply • Demand • Transshipment • We’ll use negative numbers to represent supplies and positive numbers to represent demand. • Exception: Note that in transportation problem both supply & demand figures are positive.

  7. Node types & Transshipment Formulation Node types: Supply, transshipment, demand Consider “inflow” & “outflow” to each node on the network. Objective & decision Variables? What kinds (sets) of constraints?

  8. The American Car Association ( ACA) The American Car Association ( ACA) provides a variety of travel- related services to its members, including information on vacation destinations, discount hotel reservations, emergency road assistance, and travel route planning. This last service, travel route planning, is one of its most popular services. When members of the ACA are planning to take a driving trip, they call the organizations toll-free 800 number and indicate what cities they will be traveling from and to. The ACA then determines an optimal route for traveling between these cities. The ACAs computer databases of major highways and interstates are kept up-to-date with information on construction delays and detours and estimated travel times along various segments of roadways. Members of the ACA often have different objectives in planning driving trips. Some are interested in identifying routes that minimize travel times. Others, with more leisure time on their hands, want to identify the most scenic route to their desired destination. The ACA wants to develop an automated system for identifying an optimal travel plan for its members. Suppose a travel member who wants to drive from Birmingham, Alabama to Virginia Beach, Virginia.

  9. Shortest path problem • View it as a special case of transshipment problem • Min the shipping costs of one unit from node 1 to node 11 (end node). • Node 1 supplies 1 unit, node 11 demands 1 unit, all others have zero demand or supply

  10. Shortest path formulation Objective & decision Variables? What kinds (sets) of constraints?

  11. The Northwest Petroleum Company The Northwest Petroleum Company operates an oil field and refinery in Alaska. The crude obtained from the oil field is pumped through the network of pumping sub-stations shown in Figure below to the company’s refinery located 500 miles from the oil field. The amount of oil that can flow through each of the pipelines, represented by the arcs in the network, varies due to differing pipe diameters. The numbers next to the arcs in the network indicate the maximum amount of oil that can flow through the various pipelines ( measured in thousands of barrels per hour). The company wants to determine the maximum number of barrels per hour that can flow from the oil field to the refinery.

  12. Maximal flow problem • Determine the max amount of flow that can occur from node 1 to node 6 (end node). • Max oil & gas, H2O, etc. flows • Max traffic flow (evacuations!) • Section 5.6 example; Northwest petroleum co…

  13. Minimal spanning tree problem • Determine the set of arcs that connects all the nodes at minimum cost. • By definition, given a network with n nodes, a spanning tree is a set of n-1 arcs that connects all the nodes and contains no loops. • A simple algorithm: • Select any node and call it the current subnetwork. • Add to the current subnetwork the cheapest arc that connects any node within the current subnetwork to any node not in the current subnetwork (arbitrarily break ties for the cheapest arc) Call this the current subnetwork. • Repeat step 2 until all nodes are connected • Class exercise(s): Omega Airlines • Case 5.4

  14. Minimal spanning tree example 5 D 4 A 6 12 E 15 10 B 9 8 11 7 G C F 5 8

  15. 1 5 5 3 2 6 5 3 4 4 2 6 5 8 4 6 7 6 Another formulation Although nuclear plant safety is at its highest levels, should there be an emergency, communities near nuclear plants depend on the local governments and plant operators for rapid and efficient evacuation of the public to safe areas. Duqe Power is interested in determining the optimal plan for the following network of roads and highways all heading away from the nuclear plant site, which is located at node 1. The directional arcs represent roads or highways and the circles (nodes) represent intersections. During the emergency period people (in cars, buses, etc.) will only be allowed to travel in the direction of the arrows. The numbers next to the arcs represent the maximum amount of people that can travel between nodes, measured in thousands of people per hour. Nodes 6 and 7 are the final destinations (safe areas). Develop a formulation for this problem and solve using Solver or RiskSolver or LINDO.

  16. Exercises • Suggested problems: 14, 19, 20, 31 • Interesting problems: • Omega Airlines • Check fig. = 15 • Information Consultant travels… • Group Exercise: • Student assignment case – single objective version

  17. Omega Airlines Omega Airlines has several nonstop flights between Atlanta and Los Angeles every day. The schedules of these flights are shown in the following table. Omega wants to determine the optimal way of assigning flight crews to these different flights. The company wants to ensure that the crews always return to the city from which they left each day. FAA regulations require at least one hour of rest for flight crews between flights. However, flight crews become irritated if they are forced to wait for extremely long periods of time between flights, so Omega wants to find an assignment of flight schedules that minimizes these waiting periods. Draw a network flow model for this problem & Implement the problem in a spreadsheet and solve it. What is the longest period of time that a flight crew has to wait between flights, according to your solution?

  18. Information Consultant Travels…

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